Digital ACT Practice Questions
Access 177 official-caliber practice questions. Filter by subject or difficulty to pinpoint your cognitive blind spots.
The polynomial \( P(x) = x^4 - 2x^3 - 11x^2 + 12x + 36 \) has a known root of \( x = 3 \) with multiplicity 2. What are the other two roots of the polynomial?...
A circle in the \( xy \)-plane has its center at \( (3, -2) \) and is tangent to the line \( 4x - 3y = 25 \). What is the area of the circle?...
For \( 0 \le \theta < 2\pi \), how many distinct values of \( \theta \) satisfy the equation \( \sin(2\theta) + \cos(\theta) = 0 \)?...
A hyperbola in the standard \( xy \)-coordinate plane is defined by the equation: \[ \frac{(x - 2)^2}{25} - \frac{(y + 3)^2}{36} = 1 \] What is the slope of the...
A high school academic challenge team consists of 5 seniors and 4 juniors. If a 3-person travel squad is chosen entirely at random, what is the probability that...
Given that \( \sin(\theta) = \frac{3}{5} \) and \( \theta \) lies in Quadrant II \( \left(\frac{\pi}{2} < \theta < \pi\right) \), what is the exact value of \( ...
The first term of an infinite geometric series is 18, and the sum of the series is 54. What is the common ratio \( r \) of the series?...
Matrices \( A \) and \( B \) are given by: \[ A = \begin{pmatrix} 4 & -3 \\ 2 & k \end{pmatrix}, \quad B = \begin{pmatrix} 6 & 1 \\ -2 & 3 \end{pmatrix} \] If t...
In triangle \( ABC \), side \( AB \) has length 7 meters, side \( BC \) has length 8 meters, and \( \cos(\angle B) = \frac{1}{4} \). What is the exact length, i...
For the imaginary unit \( i = \sqrt{-1} \), which of the following expressions is equivalent to: \[ \frac{5 + 2i}{3 - 4i} \]?...
Matrix \( A \) is defined as \( \begin{pmatrix} 2k & 4 \\ 3 & k-1 \end{pmatrix} \). For which of the following values of \( k \) will Matrix \( A \) have no mat...
In triangle \( \triangle PQR \), the length of \( PQ = 7 \), \( QR = 10 \), and the measure of \( \angle PQR = 120^\circ \). What is the exact length of side \(...
A quadratic function \( f(x) = ax^2 + bx + c \) has a minimum value of \( -4 \) at \( x = 1 \). If \( f(0) = -3 \), what is the value of \( a+b+c \)?...
A small, specialized library contains a collection of 6 distinct historical fiction novels and 4 distinct non-fiction biographies. A student, unaware of the spe...
If $i = \sqrt{-1}$, what is the value of the complex expression $\frac{5 - 3i}{2 + 4i}$ in standard form $a + bi$?...
A committee of $4$ students is to be randomly selected from a pool of $7$ seniors and $5$ juniors. What is the probability that the committee contains exactly $...
Given the $2 \times 2$ matrix $M = \begin{bmatrix} x & 3 \\ 4 & x - 1 \end{bmatrix}$, for what positive value of $x$ is the determinant of $M$ equal to $8$?...
What is the remainder when the polynomial $P(x) = 2x^4 - 5x^3 + 3x - 7$ is divided by $(x - 2)$?...
If $\sin(\theta) = -\frac{5}{13}$ and $\tan(\theta) > 0$, what is the exact value of $\cos(\theta)$?...
What is the sum of the infinite geometric series $18 - 12 + 8 - \frac{16}{3} + \cdots$?...
The volume of a sphere is $288\pi$ cubic units. What is the surface area, in square units, of this sphere?...
In the standard $(x,y)$-coordinate plane, the line $3x - 4y = 12$ is perpendicular to the line passing through points $(2, -1)$ and $(5, k)$. What is the value ...
If $\log_b(3) = x$ and $\log_b(5) = y$, what is the value of $\log_b(45\sqrt{b})$ expressed in terms of $x$ and $y$?...
A right circular cone and a right circular cylinder have equal heights and equal base radii. If the volume of the cylinder is $54\pi$ cubic centimeters, what is...
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